scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

scholarly journals ABSOLUTELY CONTINUOUS COMPENSATORS

2011 ◽  
Vol 14 (03) ◽  
pp. 335-351 ◽  
Author(s):  
SVANTE JANSON ◽  
SOKHNA M'BAYE ◽  
PHILIP PROTTER
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Stochastic Differential Equation ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Random Measure ◽  
Stopping Times ◽  
Poisson Random Measure ◽  
Absolutely Continuous ◽  
Strong Markov

We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Çinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation.

Buckling and Vibration of Isosceles Triangular Plates having the Two Equal Edges Clamped and the Other Edge Simply–Supported

1955 ◽  
Vol 59 (530) ◽  
pp. 151-152 ◽  
Author(s):  
Hugh L. Cox ◽  
Bertram Klein
Keyword(s):  
Differential Equation ◽  
Axial Load ◽  
Unit Length ◽  
Approximate Solutions ◽  
The Other ◽  
Thin Plates ◽  
Critical Buckling Load ◽  
Simply Supported ◽  
Governing Differential Equation ◽  
Image Position

Approximate Solutions obtained by the method of collocation are presented for the lowest critical buckling load of an isosceles triangular plate loaded as shown in Fig. 1. Also, the fundamental frequency is given. The base of the triangle is simply supported and the other equal edges are clamped. The usual assumptions regarding the bending of thin plates are made. The governing differential equation for the plate loaded as shown in Fig. 1 is1where D is the plate stiffness, N is axial load per unit length, w is deflection, positive downward, and the quantities a and h are dimensions shown in Fig.1.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

2015 ◽  
Vol 22 (04) ◽  
pp. 1550022
Author(s):  
Matteo Gregoratti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
The Other ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Proper Formulation

We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.

scholarly journals STOCHASTIC STABILIZATION OF DYNAMICAL SYSTEMS USING LÉVY NOISE

2010 ◽  
Vol 10 (04) ◽  
pp. 509-527 ◽  
Author(s):  
DAVID APPLEBAUM ◽  
MICHAILINA SIAKALLI
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Dynamical Systems ◽  
Nonlinear Differential Equation ◽  
Random Noise ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Stochastic Stabilization ◽  
Perturbed System ◽  
Exponentially Stable

We investigate the perturbation of the nonlinear differential equation [Formula: see text] by random noise terms consisting of Brownian motion and an independent Poisson random measure. We find conditions under which the perturbed system is almost surely exponentially stable and estimate the corresponding Lyapunov exponents.

scholarly journals Model of Continuous Random Cascade Processes in Financial Markets

2020 ◽  
Vol 8 ◽  
Author(s):  
Jun-ichi Maskawa ◽  
Koji Kuroda
Keyword(s):  
Differential Equation ◽  
Time Series ◽  
Stochastic Differential Equation ◽  
Financial Markets ◽  
Stock Price ◽  
Planck Equation ◽  
Cascade Model ◽  
The Other ◽  
Model Parameters ◽  
Cascade Processes

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

A Classical Stochastic Verification Theorem for Stochastic Recursive Optimization Problems

2013 ◽  
Vol 278-280 ◽  
pp. 1742-1745
Author(s):  
Shao Lin Ji ◽  
Lin Wang ◽  
Shu Zhen Yang
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Optimization Problems ◽  
Backward Stochastic Differential Equation ◽  
Verification Theorem ◽  
Recursive Optimization

In this paper, we study a stochastic recursive optimal control problem in which the system is governed by a forward-backward stochastic differential equation. Under mild assumptions, a classical stochastic verification theorem is derived.

scholarly journals Some analytic approximations for backward stochastic differential equations

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2235-2251
Author(s):  
Jasmina Ðordjevic
Keyword(s):  
Differential Equation ◽  
Iterative Method ◽  
Stochastic Differential Equation ◽  
General Type ◽  
Backward Stochastic Differential Equation ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Initial Equation ◽  
Analytic Approximations ◽  
Approximate Equations

We consider an analytic iterative method to approximate the solution of the backward stochastic differential equation of general type. More precisely, we define a sequence of approximate equations and give sufficient conditions under which the approximate solutions converge with probability one and in pth moment sense, p ? 2, to the solution of the initial equation under Lipschitz condition. The Z-algorithm for this iterative method is introduced and some examples are presented to illustrate the theory.

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